The Lagrangian/Hamiltonian encodes the dynamics of your model. As Orodruin said, you don't need both since the one is a Legendre transformation of the other [ you can always change from the 1 to the other], something that is true in classical mechanics as well.

Now for your second question, I don't really understand what caused you this confusion:
For example you want to describe a QED model- your Lagrangian will contain both photons and fermions [charged]...you could as well emit the fermions but your model would be unrealistic and boring. So you don't have much to specify about the fields...
As far as I know, the only thing that you need to define your fields with, is to state how they transform under the model (symmetry model) in consideration.

The Lagrangian/Hamiltonian encodes the dynamics of your model. As Orodruin said, you don't need both since the one is a Legendre transformation of the other [ you can always change from the 1 to the other], something that is true in classical mechanics as well.

Now for your second question, I don't really understand what caused you this confusion:
For example you want to describe a QED model- your Lagrangian will contain both photons and fermions [charged]...you could as well emit the fermions but your model would be unrealistic and boring. So you don't have much to specify about the fields...
As far as I know, the only thing that you need to define your fields with, is to state how they transform under the model (symmetry model) in consideration.

Is the Lagrangian/Hamiltonian enough to specify all possible models? For example. Is it possible for a model that needs more than the Lagrangian/Hamiltonian to specify it?

That depends on what you mean by "all possible models". The Lagrangian certainly is enough for all models in Lagrangian mechanics by definition. Of course there may be some wild theories out there not described by Lagrangian mechanics and it would be presumptuous to assume that anything can ever specify "all" models. You need a qualifier for what you consider a "possible model".

That depends on what you mean by "all possible models". The Lagrangian certainly is enough for all models in Lagrangian mechanics by definition. Of course there may be some wild theories out there not described by Lagrangian mechanics and it would be presumptuous to assume that anything can ever specify "all" models. You need a qualifier for what you consider a "possible model".

For example.. quantum mechanics and general relativity being emergence from another theory that doesn't use Lagrangian or Hamiltonian.. does this statement even makes sense? I'm asking if a theory can exist that doesn't use Lagrangian/Hamiltonian that can unite QM and GR. Any example or papers?

To be honest I don't understand Orodruin's point, neither something being not described by a Lagrangian/Hamiltonian... It's like trying to deal with something without caring about the dynamics [the Hamiltonian for example contains information about the energies; kinetic and potential]. I understand the "wild theories", but I'd be confident enough to call those theories more than just "wild".

To be honest I don't understand Orodruin's point, neither something being not described by a Lagrangian/Hamiltonian... It's like trying to deal with something without caring about the dynamics [the Hamiltonian for example contains information about the energies; kinetic and potential]. I understand the "wild theories", but I'd be confident enough to call those theories more than just "wild".

Obviously you need a hamiltonian/lagrangian to describe things in Hamilton/Lagrange formalism. Within those formalisms the hamiltonian/lagrangian is all there is by definition.

And yes, most other theories are going to be "wild" and clearly wrong. However, there will always be the possibility that there is some more fundamental type of description from which the Hamilton/Lagrange formalisms would be limits. In such a theory, there would clearly be a new notion of how dynamics appear.